共查询到20条相似文献,搜索用时 15 毫秒
1.
László Erdős Sandrine Péché José A. Ramírez Benjamin Schlein Horng‐Tzer Yau 《纯数学与应用数学通讯》2010,63(7):895-925
We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e?U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( input amssym $Bbb R$ ) with at most polynomially growing derivatives and ν(x) ≥ Ce?C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc. 相似文献
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We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure ν ij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution ν ij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk. 相似文献
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We consider ensembles of N×N Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law, as N tends to infinity. 相似文献
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Let μ be a measure with compact support, with orthonormal polynomials {p
n
} and associated reproducing kernels {K
n
}. We show that bulk universality holds in measure in {ξ: μ′(ξ) > 0}. More precisely, given ɛ, r > 0, the linear Lebesgue measure of the set {ξ: μ′(ξ) > 0} and for which
$\mathop {\sup }\limits_{\left| u \right|,\left| v \right| \leqslant r} \left| {\frac{{K_n (\xi + u/\tilde K_n (\xi ,\xi ),\xi + v/\tilde K_n (\xi ,\xi ))}}
{{K_n (\xi ,\xi )}}} \right. - \left. {\frac{{\sin \pi (u - v)}}
{{\pi (u - v)}}} \right| \geqslant \varepsilon$\mathop {\sup }\limits_{\left| u \right|,\left| v \right| \leqslant r} \left| {\frac{{K_n (\xi + u/\tilde K_n (\xi ,\xi ),\xi + v/\tilde K_n (\xi ,\xi ))}}
{{K_n (\xi ,\xi )}}} \right. - \left. {\frac{{\sin \pi (u - v)}}
{{\pi (u - v)}}} \right| \geqslant \varepsilon 相似文献
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Zhonggen SU 《Frontiers of Mathematics in China》2013,8(3):609-641
Let
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We introduce a new class of generalized Hausdorff matrices and show that their eigenvalues and corresponding eigenspaces can be obtained very easily. We also consider these matrices as operators on
p
(1p). 相似文献
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Mark Fannes Dé nes Petz 《Proceedings of the American Mathematical Society》2003,131(7):1981-1988
Let be an arbitrary self-adjoint matrix and be an (random) Wigner matrix. We show that is positive definite in the average. This partially answers a long-standing conjecture. On the basis of asymptotic freeness our result implies that is positive definite whenever the noncommutative random variables and are in free relation, with semicircular.
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M. S. Derevyagin 《Mathematical Notes》2005,77(3-4):587-591
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Yutaka Hiramine 《Designs, Codes and Cryptography》2012,62(3):279-288
Let G be a group of order mu and U a normal subgroup of G of order u. Let G/U = {U 1,U 2, . . . ,U m } be the set of cosets of U in G. We say a matrix H = [h ij ] of order k with entries from G is a quasi-generalized Hadamard matrix with respect to the cosets G/U if \({\sum_{1\le t \le k} h_{it}h_{jt}^{-1} = \lambda_{ij1}U_1+\cdots+\lambda_{ijm}U_m (\exists\lambda_{ij1},\ldots, \exists \lambda_{ijm} \in \mathbb{Z})}\) for any i ≠ j. On the other hand, in our previous article we defined a modified generalized Hadamard matrix GH(s, u, λ) over a group G, from which a TD λ (uλ, u) admitting G as a semiregular automorphism group is obtained. In this article, we present a method for combining quasi-generalized Hadamard matrices and semiregular relative difference sets to produce modified generalized Hadamard matrices. 相似文献
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We consider random matrices of the form \(H = W + \lambda V, \lambda \in {\mathbb {R}}^+\), where \(W\) is a real symmetric or complex Hermitian Wigner matrix of size \(N\) and \(V\) is a real bounded diagonal random matrix of size \(N\) with i.i.d. entries that are independent of \(W\). We assume subexponential decay of the distribution of the matrix entries of \(W\) and we choose \(\lambda \sim 1\), so that the eigenvalues of \(W\) and \(\lambda V\) are typically of the same order. Further, we assume that the density of the entries of \(V\) is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is \(\lambda _+\in {\mathbb {R}}^+\) such that the largest eigenvalues of \(H\) are in the limit of large \(N\) determined by the order statistics of \(V\) for \(\lambda >\lambda _+\). In particular, the largest eigenvalue of \(H\) has a Weibull distribution in the limit \(N\rightarrow \infty \) if \(\lambda >\lambda _+\). Moreover, for \(N\) sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for \(\lambda >\lambda _+\), while they are completely delocalized for \(\lambda <\lambda _+\). Similar results hold for the lowest eigenvalues. 相似文献
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XIE JunShan 《中国科学 数学(英文版)》2014,57(6):1221-1236
This paper focuses on the dilute real symmetric Wigner matrix Mn=1√n(aij)n×n,whose offdiagonal entries aij(1 i=j n)have mean zero and unit variance,Ea4ij=θnα(θ0)and the fifth moments of aij satisfy a Lindeberg type condition.When the dilute parameter 0α13and the test function satisfies some regular conditions,it proves that the centered linear eigenvalue statistics of Mn obey the central limit theorem. 相似文献
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Maxim Derevyagin 《Journal of Mathematical Analysis and Applications》2011,384(2):444-452
Some inverse problems for semi-infinite periodic generalized Jacobi matrices are considered. In particular, a generalization of the Abel criterion is presented. The approach is based on the fact that the solvability of the Pell-Abel equation is equivalent to the existence of a certainly normalized J-unitary 2×2-matrix polynomial (the monodromy matrix). 相似文献
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Periodica Mathematica Hungarica - Studying eigenvalues of square matrices is a traditional and fundamental direction in linear algebra. Quaternion matrices constitute an important and extensively... 相似文献
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Inversion formulas for infinite generalized Toeplitz matrices 总被引:3,自引:0,他引:3
Inversion formulas are obtained for a certain class of infinite matrices that possess displacement structure similar to that of finite block Toeplitz matrices. Consequences are symmetric inversion formulas for matrix-valued singular integral operators and infinite Toeplitz plus Hankel matrices. 相似文献
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